Top

Designs, Codes and Cryptography

Published in:

16-08-2016

The BEL-rank of finite semifields

Authors: Michel Lavrauw, John Sheekey

Published in: Designs, Codes and Cryptography | Issue 3/2017

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this article we introduce the notion of the BEL-rank of a finite semifield, prove that it is an invariant for the isotopism classes, and give geometric and algebraic interpretations of this new invariant. Moreover, we describe an efficient method for calculating the BEL-rank, and present computational results for all known small semifields.

previous article Doubly resolvable Steiner quadruple systems and related designs

next article Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

Albert A.A.: Finite division algebras and finite planes. In: Proceedings of Symposia in Applied Mathematics, vol. 10, pp. 53–70. American Mathematical Society, Providence (1960).

André J.: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156–186 (1954).MathSciNetCrossRefMATH

Ball S., Lavrauw M.: On the Hughes–Kleinfeld and Knuth’s semifields two-dimensional over a weak nucleus. Des. Codes Cryptogr. 44, 63–67 (2007).MathSciNetCrossRefMATH

Ball S., Ebert G., Lavrauw M.: A geometric construction of finite semifields. J. Algebra 311, 117–129 (2007).MathSciNetCrossRefMATH

Cardinali I., Polverino O., Trombetti R.: Semifield planes of order \(q^4\) with kernel \(F_{q^2}\) and center \(F_q\). Eur. J. Comb. 27, 940–961 (2006).MathSciNetCrossRefMATH

Dempwolff U.: Semifield planes of order 81. J. Geom. 89, 1–16 (2008).MathSciNetCrossRefMATH

Hou X., Özbudak F., Zhou Y.: Switchings of semifield multiplications. arXiv:1406.1067v1.

Kantor W.M.: Finite Semifields. Finite Geometries, Groups, and Computation, pp. 103–114. Walter de Gruyter GmbH & Co. KG, Berlin (2006).

Knuth D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965).MathSciNetCrossRefMATH

10.

Lavrauw M.: Scattered spaces with respect to spreads, and eggs in finite projective spaces. Thesis (Dr.), Technische Universiteit Eindhoven (2001).

11.

Lavrauw M.: On the isotopism classes of finite semifields. Finite Fields Appl. 14, 897–910 (2008).MathSciNetCrossRefMATH

12.

Lavrauw M.: Finite semifields with a large nucleus and higher secant varieties to Segre varieties. Adv. Geom. 11, 399–410 (2011).MathSciNetCrossRefMATH

13.

Lavrauw M.: Finite semifields and nonsingular tensors. Des. Codes Cryptogr. 68, 205–227 (2013).MathSciNetCrossRefMATH

14.

Lavrauw M., Polverino O.: Finite semifields. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry. NOVA Academic Publishers, New York (2011).

15.

Lavrauw M., Sheekey J.: On BEL-configurations and finite semifields. Des. Codes Cryptrogr. 78(3), 583–603 (2016).MathSciNetCrossRefMATH

16.

Lavrauw M., Van de Voorde G.: Field reduction and linear sets in finite geometry. Topics in finite fields. Contemp. Math. 632, 271–293 (2015).CrossRefMATH

17.

Marino G., Polverino O.: On the nuclei of a finite semifield. Contemp. Math. 579, 123–141 (2012).MathSciNetCrossRefMATH

18.

Marino G., Polverino O., Trombetti R.: On \(\mathbb{F}_q\)-linear sets of PG\((3,q^3)\) and semifields. J. Comb. Theory Ser. A 114, 769–788 (2007).CrossRefMATH

19.

Menichetti G.: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field. J. Algebra 47, 400–410 (1977).MathSciNetCrossRefMATH

20.

Rúa I.F., Combarro E.F., Ranilla J.: Classification of semifields of order 64. J. Algebra 322, 4011–4029 (2009).MathSciNetCrossRefMATH

21.

Rúa I.F., Combarro E.F., Ranilla J.: New advances in the computational exploration of semifields. Int. J. Comput. Math. 88, 1990–2000 (2011).MathSciNetCrossRefMATH

22.

Rúa I.F., Combarro E.F., Ranilla J.: Determination of division algebras with 243 elements. Finite Fields Appl. 18, 1148–1155 (2012).MathSciNetCrossRefMATH

Title: The BEL-rank of finite semifields
Authors: Michel Lavrauw
John Sheekey
Publication date: 16-08-2016
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 3/2017
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-016-0270-z

Springer Professional

The BEL-rank of finite semifields

Abstract

Premium Partner

Springer Professional

Abstract

Please log in to get access to your license.

Other articles of this Issue 3/2017

Near-complete external difference families

Classification of partial difference sets in Abelian groups of order

Higher order differentiation over finite fields with applications to generalising the cube attack

Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

On the power of rewinding simulators in functional encryption

Premium Partner